Key takeaway: A nonlinear function is a function whose rate of change is not constant, so its graph is a curve instead of a straight line.
Grade 88.F.A.3Functions and Graphs
Definition of Nonlinear Function
A nonlinear function changes at a different pace in different places. The area of a square is the classic case: double the side and the area quadruples, so the bigger the square gets, the faster its area grows with each added inch of side.
You can spot one in three ways. In a table, the output differences are unequal for equal input steps. On a graph, the points bend instead of lining up. In an equation, the input may be squared or cubed, sit in a denominator, or serve as an exponent. Growth in savings with interest, the path of a thrown ball and the speed of a falling stone are all nonlinear. At this level, a nonlinear function is mainly recognized, compared with linear ones, and described in words.
Formula and notation
Typical rules include
\[A(s) = s^{2}, \qquad f(x) = \frac{12}{x}, \qquad g(x) = 2^{x}\]
Each fails the test \(f(x) = mx + b\). In \(A(s) = s^{2}\), the rate of change over an interval depends on where the interval is.
Key facts
- Table differences are unequal for equal input steps.
- The graph is a curve, or a path that bends or changes direction.
- The average rate of change depends on which interval you pick.
- Squares, cubes, reciprocals and exponentials all give nonlinear rules.
Nonlinear Function example
A banner maker records the area of square banners: side 2 inches gives area 4, side 4 gives 16, side 6 gives 36 and side 8 gives 64. Is area a linear or nonlinear function of the side?
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The input steps are equal: 2, 4, 6, 8, each increasing by 2.
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The output steps are \(16 – 4 = 12\), \(36 – 16 = 20\) and \(64 – 36 = 28\).
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The rates are \(\frac{12}{2} = 6\), \(\frac{20}{2} = 10\) and \(\frac{28}{2} = 14\). They are not equal.
The rate keeps changing, so area is a nonlinear function of the side: \(A = s^{2}\).
Common mistake
Watch out
Hal sees that a 2-inch square has area 4 and assumes a 4-inch square has area 8, because “doubling the side doubles the area.”
Why it is wrong
Area depends on two lengths, so doubling the side multiplies the area by 4. The 4-inch square has area 16. Assuming proportional growth treats the curve as a line and underestimates every larger value.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Functions and Graphs